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Evaluation of manoeuvring coefficients of a self-propelled ship using a Blade Element Momentum propeller model coupled to a Reynolds averaged Navier Stokes flow solver

Alexander Phillips1, Stephen R. Turnock1, Maaten Furlong2 1 School of Engineering Sciences, University of Southampton

2National Oceanography Centre, Southampton Abstract The use of an unsteady Computational Fluid Dynamic analysis of the manoeuvring

performance of a self-propelled ship requires a large computational resource that

restricts its use as part of a ship design process. A method is presented that

significantly reduces computational cost by coupling a Blade Element Momentum

Theory (BEMT) propeller model with the solution of the Reynolds Averaged Navier

Stokes (RANS) equations. The approach allows the determination of manoeuvring

coefficients for a self-propelled ship travelling straight ahead, at a drift angle and for

differing rudder angles. The swept volume of the propeller is divided into discrete

annuli for which the axial and tangential momentum changes of the fluid passing

through the propeller are balanced with the blade element performance of each

propeller section. Such an approach allows the interaction effects between hull,

propeller and rudder to be captured. Results are presented for the fully appended

model scale self propelled KVLCC2 hull form under going static rudder and static

drift tests at a Reynolds number of 4.6x106 acting at the ship self propulsion point. All

computations were carried out on a typical workstation using a hybrid finite volume

mesh size of 2.1x106 elements. The computational uncertainty is typically 2-3% for

side force and yaw moment.

Keywords: ship manoeuvring; self propulsion, CFD, validation, rudder-

propeller interaction, self-propulsion

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Nomenclature a Axial flow factor a’ Circumferential flow factor B Number of blades BT Breadth of tank (m) C Blade chord (m) CD Drag coefficient CL Lift coefficient D Drag (N) DT Depth of tank (m) J Advance coefficient K Goldstein correction factor KQ Torque coefficient KT Thrust coefficient L Lift (N) Lbp Length between perpendiculars (m) n Revolutions per second (s-1) N Yaw moment (Nm) P Pressure (Pa) Q Torque (Nm) R Local radius (m) Rk Mesh refinement ratio R Resistance (N) RN Reynolds number Rx Rudder x force (N) Ry Rudder y force (N) t Thrust deduction factor T Thrust (N) ui Cartesian velocity components (m/s) v Ship sway velocity (m/s) Va Propeller advance velocity (m/s) wt nominal wake fraction wt’ Local nominal wake fraction xi Cartesian co-ordinates (m) X Longitudinal force (N) X/D Longitudinal separation between rudder and propeller /

propeller diameter Y Transverse force (N) α Incidence angle ∆KT Increase in thrust coefficient due to presence of rudder ∆ wt Increase in wake fraction due to the presence of the rudder η Efficiency ηi Ideal efficiency r Fluid density (kg/m3) φ Hydrodynamic pitch angle (deg) ψ Undisturbed flow angle (deg) Ω Rotational velocity (s-1)

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1 Introduction

The design assessment of the manoeuvring performance of a ship requires the

evaluation of between 20 -30 derivative coefficients, ed. Comstock (1967). Such

coefficients depend on the interaction between the forces and moments generated on

the hull and rudder in the presence of the propeller, Molland and Turnock (2007). In

order to accurately assess the forces acting on a manoeuvring vessel the interaction of

the hull, propeller and rudder must be considered in the calculation of global forces. It

is now possible to compute the interaction using an unsteady computational fluid

dynamics (CFD) solver that models the full flow regime around the hull, propeller and

rudder, ITTC, (2008). However, the necessary time step and mesh refinement, which

are driven primarily by the need to capture the flow field around the propeller, results

in a very large computational cost. The method proposed in this paper makes use of

the flow integrating effect of the propeller which generates an accelerated and swirled

onset flow onto the rudder while the rudder acts to block and divert the flow through

the propeller, Turnock(1993). As long as the radial variation in axial and tangential

momentum (including hull and rudder interaction effects) generated by the propeller

are included, then the influence of the unsteady propeller flow can be removed and

‘steady’ calculations performed to evaluate the manoeuvring coefficients.

Table 1 presents the hierarchy of numerical methods for modelling propellers

ranging in complexity and accuracy, Bertram (2000), Breslin and Anderson (1994).

The actuator disc model based on momentum theory was implemented within a

Reynolds Averaged Navier Stokes (RANS) simulation of the flow about a hull by

Schetz and Favin(1977). Momentum theory can be adapted to run in conjunction with

a RANS simulation where the predicted thrust and torque are implemented in the

RANS simulation as a series of momentum sources distributed over the propeller disc.

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Table 1: Numerical methods for modelling propellers

Method Description Momentum Theory The propeller is modelled as an actuator

disc over which there is a instantaneous pressure change, resulting in a thrust acting at the disc. The thrust, torque and delivered power are attributed to changes in the fluid velocity within the slipstream surrounding the disc, Rankine(1865), Froude (1889), and Froude (1911).

Blade Element Theory The forces and moments acting on the blade are derived from a number of independent slices represented as a 2-D aerofoils at an angle of attack to the fluid flow. Lift and drag information for the slices must be provided a priori and the induced velocities in the fluid due to the action of the propeller are not accounted for, Froude (1878) .

Blade Element Momentum Theory By combining Momentum theory with blade element theory, O’brian (1969), the induced velocity field can be found around the 2D-sections. Corrections have been presented to account for finite number of blades and strong curvature effects.

Lifting-Line Method The propeller blades are represented by lifting lines, which have a varying circulation as a function of radius, this approach is unable to capture stall behaviour, Lerbs (1925) .

Lifting surface Method The propeller blade is represented as an infinitely thin surface fitted to the blade camber line. A distribution of vorticity is applied in the spanwise and chordwise directions, Pien (1961).

Panel Method Panel methods extend the lifting surface method to account for blade thickness and the hub still by representing the surface of the blade by a finite number of vortex panels, Kerwin (1987).

Reynolds Averaged Navier Stokes Full 3D viscous flow field modelled using a finite volume or finite element approach to solve the averaged flow field, Adbel-Maksoud et al. (1998).

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The distribution of the sources may be uniform or based on radial distributions

such as presented by Hough and Ordway (1965). This approach has subsequently

been used by others such as Stern et al. (1988), Simonsen (2000), Simonsen and Stern

(2005), Carrica et al. (2008) and Miller (2008), to represent the action of the propeller

in various marine applications . Since this body force distribution is prescribed a

priori the influence of the hull on the propeller and the rudder on the propeller is not

captured by this method. More complex vortex based (lifting line) numerical methods

have also been coupled with RANS simulations, Han et al. (2007) , however, these

methods are limited by potential theory that do not for example, identify propeller

sectional stall or Reynold’s number (scaling) effects.

Full RANS based simulation of marine propellers have been shown to be

applicable for propeller design purposes, Rhee and Joshi (2005), and complete RANS

simulations of a hull and propulsor, which use sliding interfaces to model the

propeller rotation, have been performed, Taylor et al. (1998) and Lubke (2007).

Mueller et al. (2006), used a RANS code to model the self-propelled flow around an

appended Ro-Ro Twin Screw ferry travelling straight ahead. Their comparison of an

explicit modelling of the full propeller through steady state and transient RANS

simulation with use of a coupled blade element model of the propeller with the RANS

flow over the hull showed a 90% reduction in computational effort, with only a small

reduction in accuracy.

In order to reduce the significant computational cost associated with modelling

the full flow field around a self propelled ship model, the RANS simulation of the

flow around the manoeuvring hull form is coupled with an external Blade Element

Momentum Theory (BEMT) code, Molland and Turnock (1996) developed to model

propeller-rudder interaction. The implementation of this approach is described and

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results are given for the prediction of the self-propulsion and manoeuvring

coefficients for the KVLCC2 hull form, Van et al, (1998), Kim et al (2001).

2 Blade Element Momentum Theory Blade element momentum theory (BEMT) is used in the design of wind

turbines, Burton et al.(2001) and Mikkelsen (2003), tidal turbines , Batten et al.

(2006) and Nicholls-Lee and Turnock (2007), and its importance for ship propeller

analysis is discussed by Benini (2004). An advantage of BEMT theory over advanced

methods is that it allows the lift and drag properties of the 2D sections representing

the blade to be tuned to the local operating Reynolds number incorporating viscous

effects such as stall or the effect of laminar separation at low Reynolds No. For

turbulent flows, stall effects are difficult to capture using a RANS approach with

results strongly dependent on turbulence closure models, while capturing laminar or

transitional propeller sectional flow features can not be achieved using a conventional

RANS turbulence closure.

Figure 1: Actuator Disc Theory

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BEMT combines the two dimensional action of the blade from blade element

theory with momentum changes in the fluid from momentum theory to determine the

effective angle of attack of each section and hence thrust and torque components of

each section. The flow through an annulus of radius r and thickness dr at the propeller

disc is considered. It can be shown that the increment of axial velocity at the disc is

half that of downstream, see Figure 1, the thrust and torque acting on a length of blade

dr can also be deduced. The thrust can be written as: -

)1(4 2 akarVdrdT

+= rπ (1),

where a is the axial inflow factor and k is the Goldstein correction to account

for the propeller having a finite number of blades, Goldstein (1929). Similarly the

torque can be written as: -

)1('4 3 aVkardrdQ

+Ω= rπ , (2)

where Ω is the angular velocity of the propeller and a’ is the circumferential inflow

factor. The ideal efficiency ηi is found as from (1) and (2) as: -

aa

drdQdrdTV

i +−

=Ω

=1

'1η . (3)

Non dimensionalising equations (1) and (2) in terms of KT, KQ and J : -

)1(2 axkaJdr

dKT += π , (4)

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)1('21 32 akaJx

drdKQ += π . (5)

The local lift and drag acting on the 2D blade section is given by:-

)(21 2 αr lCNCU

drdL

= , (6)

)(21 2 αr dCNCU

drdD

= , (7)

where N is the number of blades, C is the blade chord and the lift and drag

coefficients Cl and Cd depend on the angle of attack α and are determined from

experimental tests or numerically for the 2D section1. The section lift and drag, see

Figure 2, are resolved to give the annulus torque and thrust: -

)tantan1(cos γφφ −=drdL

drdT , (8)

)tan(tancos γφφ +=drdLr

drdQ . (9)

Combining equations (8) and (9) gives the local section efficiency: -

)tan(tan

γφψη+

= . (10)

1 In this case 0107.01.0 0 == dl CddC α with section lift values calculated for zero lift incidence and drag increased as stall approached.

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The ideal efficiency from momentum theory and the local efficiency from

blade element theory allows the axial and circumferential inflow factors a and a’ to be

found at each section dr along the blade: -

)1(1' aa i +−= η , (11)

Ψ+

−=

2tan11

ηη

η

i

ia . (12)

An iterative approach is used to determine the unknown sectional angle of attack and

hence actual inflow factors a and a’ by initially assuming α and that Cd=0, hence γ=0

and η=ηi.

Figure 2: Blade Element Theory

3 Reynolds Averaged Navier Stokes (RANS) formulation

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The motion of the fluid is modelled using the incompressible (13), isothermal

Reynolds Averaged Navier Stokes (RANS) equations (14) in order to determine the

Cartesian flow (ui = u, v, w) and pressure (p) field of the water around the ship hull

and rudder: -

0=∂∂

i

i

xU , (13)

ii

ji

i

j

j

i

iii

jii fxuu

xU

xU

xxP

xUU

tU

+∂

′′∂−

∂

∂+

∂∂

∂∂

+∂∂

−=∂

∂+

∂∂

νr1 . (14)

The RANS equations are implemented in the commercial CFD code ANSYS

CFX 11 (CFX), ANSYS CFX (2006). The governing equations are discretised using

the finite volume method. The high-resolution advection scheme was applied for the

results presented which varies between first and second order accuracy depending on

spatial gradient. For a scalar quantity φ the advection scheme is written in the form

Rbupip .φφφ ∇+= , (16)

where φip is the value at the integration point, φup is the value at the upwind node and

R is the vector from the upwind node to the integration point. The model reverts to

first order when b=0 and is a second order upwind biased scheme for b=1. The high

resolution scheme calculates b using a similar approach to that of Barth and Jeperson

(1989), which aims to maintain b locally to be as close to one as possible without

introducing local oscillations. Collocated (non-staggered) grids are used for all

transport equations, and pressure velocity coupling is achieved using an interpolation

scheme based on that proposed by Rhie and Chow (1982). Gradients are computed at

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integration points using tri-linear shape functions defined in ANSYS CFX (2006).

The linear set of equations that arise by applying the Finite Volume Method to all

elements in the domain are discrete conservation equations. The system of equations

is solved using a coupled solver and a multigrid approach.

The Shear Stress Transport (SST), Menter (1994), turbulence closure model

was selected for this study since previous investigations have shown that it is better

able to replicate the flow around ship hull forms than either zero equation models or

the k-ε model, notably in capturing hooks in the wake contours at the propeller plane,

Larsson et al. (2003).

5 Coupled RANS BEMT Method

Within the RANS mesh the propeller is represented as a cylindrical sub-

domain with a diameter, D, equal to that of the propeller and a length of 0.15D.The

sub domain is divided into a series of ten annuli corresponding to ten radial slices (dr)

along the blade. The appropriate momentum source terms from (4) and (5) are then

applied over the sub-domain in cylindrical co-ordinates to represent the axial and

tangential influence of the propeller.

The following procedure is adopted in order to calculate the propeller

performance and replicate it in the RANS simulations.

1. An initial converged stage of the RANS simulation (RMS Residuals < 1E − 5)

of flow past the hull is performed, with the propeller domain body force terms

set to zero.

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2. The local nominal wake fraction, wT’, is determined for each annulus by

calculating the average circumferential mean velocity at the corresponding

annuli.

,121 2

0

' ∫

−=

π

θπ

rdVU

rw

aT (15)

where U is the axial velocity at a given r and θ. This captures the influence of

the hull and rudder on the flow through and across the propeller disc. This

calculation is written as a user specified Fortran module that exports the set of

local axial wake fractions to the BEMT code.

3. The BEMT code iterates to find the thrust (dKT ) and torque (dKQ) for the ten

annuli based on ship speed, the local nominal wake fraction and the propeller

rpm and applying (4) through (10). A converged solution is deemed to have

occurred when the difference in α is less than 1% . This phase of analysis

adds a negligible overhead to the overall computational cost.

4. The local thrust and torque derived by the BEMT code are assumed to act

uniformly over the annulus corresponding to each radial slice. The thrust is

converted to axial momentum sources (momentum/time) distributed over the

annuli by dividing the force by the volume of annuli. The torque is converted

to tangential momentum sources by dividing the torque by the average radius

of the annulus and the volume of the annulus.

5. These momentum sources are then returned to the RANS solver by the Fortran

Module which distributes them equally over the cell within the axial length of

the propeller disc.

6. The RANS simulation is then restarted from the naked hull solution but now

with the additional momentum sources. The final solution is assumed to have

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converged when the root mean square residual is less than 1E − 5. Typically

the computational cost of this second phase of the RANS simulation adds a

further 30%.

It should be noted that the procedure discussed above calculates the propeller

inflow conditions based on the nominal wake field, i.e the wake field without the

presence of the propeller. In practice when a propeller is operating in the wake of a

ship the total velocity field is the sum of the nominal wake field, the propeller induced

velocities and interaction velocities due to the complex interaction between the hull

and propeller, Carlton (2007). Thus ideally the input to the propeller model would be

the effective wake field which is the sum of nominal wake and induced velocities. It is

possible to find the effective wake field by repeating the process from steps 2 through

6 to find the total velocity field then subtracting the propeller induced velocities

calculated from the BEMT code, see Phillips et al. (2008). For the particular

geometry investigated only very small changes occur and this additional iterative loop

was not included.

6 Evaluation of the coupled BEMT-RANS using the KVLCC2 hull

The KRISO Very Large Crude Carrier 2 (KVLCC2), see Figure 3 and Table 2

for hull particulars, is a well documented experimental test case for CFD code

evaluation, Larsson et al. (2003), Hino (2005) and Stern and Agdrup (2008). The

KVLCC2 was designed by the Korean Institute of Ships and Ocean Engineering

(KRISO now MOERI) and is representative of full bodied ships (CB = 0.81) single

screw tankers. The hull form incorporates a bulbous bow and a transom stern.

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Figure 3: KVLCC2 Hull Form

Table 2: Principal Dimensions of the KVLCC2 Model and Propeller

Dimension Full-Scale Model-Scale Scale 1.00 58.000 LPP(m) 320.0 5.5172 Bwl(m) 58.0 1.0000 D (m) 30.0 0.5172 T (m) 20.8 0.3586 ∇ (m3) 312622 1.6023 Type FP FP No. Blades 4 4 D(m) 9.86 0.170 P/D(0.7R) 0.721 0.721 Ae/Ao 0.431 0.431 Rotation Right Hand Right Hand Hub Ratio 0.155 0.155

As part of the SIMMAN 2008 Workshop on Verification and Validation of

Ship Manoeuvring Simulation Methods model tests where performed at the MOERI

test tank (200m long x 16m wide x 7 deep) on a 1/58.0 scale self propelled model, at a

range of speeds, drift angles and rudder angles. The model was fitted with a propeller

and a semi-balanced rudder based on a NACA0018 section with an area of 0.0654m2

and a geometric aspect ratio of 1.55.

A series of static drift, static rudder, pure sway planar motion mechanism

(PMM) tests and pure yaw PMM tests where performed with the propeller operating

at the ship propulsion point (515 rpm) at a model speed of 1.047m/s corresponding to

15.5 knots full scale (Fn=0.14 and model scale Rn = 4.6x106).

Figure 4 demonstrates the application of the BEMT when predicting the given

open water performance of the model propeller, provided by MOERI with the

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performance predicted by BEMT. For the effective advance speed of interest for this

work (nominal J =0.35) the agreement for KT and KQ was excellent, with a difference

of less than 1%. It should be noted that the propeller diameter based Reynolds

Number (Rn=VD/ν) is 150,000 and as such a full turbulent RANS propeller

simulation would not be appropriate. Table 3 summarises the computational

parameters adopted.

Figure 4: Comparison of Propeller Characteristics in Open Water. Experimental data made available as part of the SIMMAN workshop, Stern F., Agdrup K., editors 2008

Table 3: Computational model

Parameter Setting Computing 64-bit desktop pc 4GB of RAM Mesh Type Unstructured –hybrid (tetrahedra/prism) Turbulence Model Shear Stress Transport [40] Advection Scheme CFX High Resolution Convergence Control RMS residual <10-5

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6.1 Mesh Definition

A hybrid finite volume unstructured mesh was built with the meshing tool

ANSYS ICEM V11, using tetrahedra in the far field and inflated prisms elements

around the hull with a first element thickness equating to a y+ = 30, with 10 to 15

elements used to capture the boundary layer of both hull and rudder. Separate meshes

where produced for each rudder angle using a representation of the horn rudder with

sealed gaps between the movable and fixed part of the rudder, see Figure 5.

Figure 5: Mesh cut plane through longitudinal centreline of ship

6.2 Boundary Conditions

The solution of the RANS equations requires a series of appropriate boundary

conditions to be defined. The hull is modelled using a no-slip wall condition. Based

on previous experience a Dirichlet inlet condition, one body length upstream of the

hull is defined where the inlet velocity and turbulence are prescribed explicitly. The

model scale velocity is replicated in the CFD analyses; and inlet turbulence is set at

the default value of 5%. A mass flow outlet is positioned 3 body lengths downstream

of the hull.

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The influence of tank size on hydrodynamic derivatives determined from CFD

simulations has been demonstrated by Brogali et al. (2006), thus to replicate the test

conditions free slip wall conditions are placed at the locations of the floor and sides of

the tank (16m wide x 7m deep) to enable direct comparison with the experimental

results without having to account for blockage effects (DT /T = 19.5 and L/BT =

0.345). The influence of free surface is not included in these simulations due to the

increase in computational cost, and the free surface is modelled with a symmetry

plane. The Froude number is sufficiently low, Fn=0.14, that this is not expected to

have a large effect.

6.3 Mesh Sensitivity

An uncertainty assessment has been performed based on the methodology

presented by Stern et al. (1999). While not directly applicable to a hybrid mesh, it is

assumed to be a suitable approach when using a hybrid meshing strategy where the

mesh in the boundary layer is systematically refined. Table 4 shows the results of

mesh sensitivity study for the self propelled case with the rudder at 10o using a

refinement ratio of rk = √2 with the finest mesh having 2.1x106 elements, within the

boundary layer modification of the first later thickness modified the y+ value from 30

on the finest mesh to 60 on the coarsest.

Computational uncertainty was found to be 2-3% for side force and yaw

moment but much larger at 15% for resistance. Previous CFD workshops highlight

the difficulties in accurate prediction of straight line resistance, with large uncertainty

and comparison errors common between calculated and experimental drag unless

significantly larger meshes (10M+ elements) are used Hino ed. (2005). Thus a mesh

density of 2M cells proves inadequate to achieve a fully mesh independent solution

capturing all aspects for self propulsion and propeller design calculations.

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Nonetheless, a good level of understanding of the global forces and moments required

for manoeuvring coefficients can be obtained with this level of mesh resolution.

Table 4: Uncertainty Analysis – Self Propulsion – Rudder at 10o

Exp.

(D) Fine (SG)

Medium Coarse UG (%SG)

E (%D) Uv (%D)

Longitudinal Force X(N)

-11.05 -11.74 -12.60 -13.82 17.50 -6.28 18.77

Transverse Force, Y (N)

6.79 7.6 7.51 7.33 1.35 -11.89 2.92

Yaw Moment, N (Nm)

-19.47 -18.75 -18.70 -18.35 0.49 3.70 2.54

Thrust, T (N)

10.46 12.53 12.37 12.08 1.57 -19.79 3.13

Rudder X Force, Rx (N)

-2.02 -1.83 -1.89 -1.94 - 9.39 -

Rudder Y Force, Ry (N)

4.32 4.94 4.99 4.88 - -14.49 -

7 Results 7.1 Hull-Propeller-Rudder Interaction

Figure 6 shows the spanwise variation of dKT and dKQ along the blade due to

the local nominal wake fraction (wT’), the average nominal wake fraction (wT) over

the propeller disc was calculated at 0.467 compared with 0.443 derived

experimentally.

The momentum terms used to replicate the action of the propeller in the

RANS simulation lead to a thrust deduction factor, t=(T-R)/T, of 0.236 compared to

the value of 0.190 derived experimentally. Figures 7 and 8 illustrate the influence of

the propeller on the hull surface pressure and skin friction distribution respectively,

for the without rudder case. Note, the propeller model introduces asymmetry in the

flow by replicating the swirl effect of the propeller, consequently the illustrated

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starboard view shown differs slightly to the port side of the vessel. The local

acceleration of the flow due to the propeller leads to a reduction in local pressure

coefficient, Cp=(P-P0)/(1/2rU02), and an increase in the skin friction coefficient

Cf=τW/(1/2rU02). These effects are concentrated at the stern of the vessel, diminishing

as the parallel mid-body is approached.

Figure 6: Variation in nominal wT , dKT and dKQ along blade radius

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Figure 7: Comparison of streamlines passing through the propeller disc for the appended hull, no propeller model (top), propeller model on (bottom)

The action of the propeller accelerates the flow and induces a swirl

component. This travels downstream where it flows onto the rudder significantly

changing the flow around the rudder without the propeller, see Figure 9. The net result

of the propeller action is an increase in the velocity and an effective angle of

incidence, leading to an increase in rudder drag and the production of rudder lift, with

the rudder at zero incidence.

The presence of the rudder modifies the flow upstream into the propeller

influencing the performance of the propeller, see Table 5. Blockage from the rudder

reduces the flow velocity into the propeller, increasing the nominal mean wake

fraction , wT by 0.018, resulting in an increase in the thrust and torque coefficients.

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The change in the local wake fraction and the drag from the rudder modifies the rpm

required at the model propulsion point from 552 rpm to 542rpm. Comparing the thrust

coefficient with and without the action of the propeller results in a ∆KT = 0.07 at a

rudder separation X/D = 0.28. This compares well with the wind tunnel experiments

by Molland and Turnock (2007) which derived a ∆KT = 0.058 at a X/D = 0.3 for a

similar rudder and propeller thrust loading.

7.2 Global Forces

The global loads acting on the vessel are non-dimensionalised by the length of

the vehicle (L) the velocity of the vehicle (V) and the density of the fluid (r), a prime

symbol is used to signify the non dimensional form for example: -

Vvv =' ,

222/1'

VLYY

r= ,

232/1'

VLNN

r= .(15)

The axis system is described in Figure 10. The matching set of experiments

were performed with the vessel restrained in roll but free to heave and pitch, however,

to reduce simulation time the CFD simulations have assumed the vessel is fixed in

heave and pitch at the quoted mean draught and level keel. For the zero drift, zero

rudder angle case the force in the z direction was 253N downwards, which

corresponds to a sinkage of 5.1mm equating to 1% of the draft.

Figures 11 illustrate the variation of global forces with variation in drift angle.

The influence of drift angle on global loads is well captured even at larger amplitude

drift angles outside the linear region.

Prediction of the rudder forces is dependent on the rudder inflow conditions

which are dominated by the action of the hull and the propeller. Thus to accurately

capture the rudder forces the flow in the stern of the vessel needs to be captured with

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a high level of accuracy, to ensure the correct flow into the propeller and then across

the rudder. Small over predictions in the thrust generated by the propeller will lead to

an increased inflow velocity which will then cause an over prediction of rudder force.

This is seen in Figure 12, where the predicted propeller thrust is approximately 20%

higher than the experimental result, leading to over prediction of the global side force

and yawing moment which are dominated by the rudder loads. It should be noted that

only 13500 surface mesh elements were used to define the rudder, surrounded by a

mesh of the order 200,000 cells in the vicinity of the rudder. Work by Date and

Turnock (2002) indicate values of 5-20M cells are required to fully resolve the rudder

force.

These simulations provide good initial estimates of the manoeuvring

coefficients for the appended KVLCC2 hullform. The results would benefit from finer

mesh resolution in the boundary layer region, resolving into the viscous sub layer

would remove uncertainty resulting in using wall functions, finer resolution in the

region of the bilge vortices would improve the both the prediction of the hull surface

pressure and prediction of the propeller inflow, while further mesh density around the

rudder and rudder tip vortex would improve prediction of the rudder forces.

The coupled RANS-BEMT approach circumferentially averages propeller

inflow for a series of annuli. This fails to capture the circumferential variation in

blade loading and consequently propeller side force. A more complete approach

would be to subdivide the propeller disc into a series of discrete zones in both the

radial and circumferential direction. Such an approach would add negligible

computational cost but provide a more complete representation of the influence of the

propeller within the RANS simulation, Phillips et al. (2008).

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Figure 8: Axis System

Figure 9: Influence of Drift Angle on Force Coefficients at a Model speed of 1.047m/s Experimental data made available as part of the SIMMAN workshop, Stern F., Agdrup K., editors

2008

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Figure 10: Influence of Rudder Angle on Force Coefficients at a Model speed of 1.047m/s. Experimental data made available as part of the SIMMAN workshop, Stern F., Agdrup K., editors 2008

8 Conclusions

A method of coupling a Blade Element Momentum Theory code for marine

propellers with the commercial RANS code ANSYS CFX is presented. As a

demonstration of the effectiveness of the approach a coupled CFD-BEMT simulation

of the flow around the model scale KVLCC2 tanker was able to predict the global

forces and moments acting on a vessel undergoing self propelled steady state

manoeuvres with good agreement. This was with a computational uncertainty of 2-3%

for manoeuvring side force and yaw moment. The method captures the changes in

propulsive force associated with the downstream rudder and increase in rudder

sideforce associated with the acceleration of the propeller race. It is estimated that

computational cost of running a propeller model based approach is 1/10th of the cost

of considering the full transient flow field with an unsteady BEMT. A one to two

25

order of magnitude reduction in timestep is likely to be required to fully model the

propeller in RANS, Mueller et al, (2006).

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